COMPACT COMPOSITION OPERATORS
ON BERGMAN SPACES OF THE UNIT BALL
KEHE ZHU
A BSTRACT. Under a mild condition we show that a composition operator Cϕ is compact on the Bergman space Apα of the open unit ball in Cn
if and only if (1 − |z|)/(1 − |ϕ(z)|) → 0 as |z| → 1− .
1. I NTRODUCTION
For any positive integer n we let
Cn = C × · · · × C
denote the n-dimensional complex Euclidean space. For any two points
z = (z1 , · · · , zn ) and w = (w1 , · · · , wn ) in Cn we write
hz, wi = z1 w1 + · · · + zn wn ,
and
p
|z| = |z1 |2 + · · · + |zn |2 .
The open unit ball in Cn is the set
Bn = {z ∈ Cn : |z| < 1}.
The space of holomorphic functions in Bn will be denoted by H(Bn ).
Let dv be Lebesgue volume measure on Bn , normalized so that v(Bn ) =
1. For any α > −1 we let
dvα (z) = cα (1 − |z|2 )α dv(z),
where cα is a positive constant chosen so that vα (Bn ) = 1. The weighted
Bergman space Apα , where p > 0, consists of functions f ∈ H(Bn ) such
that
Z
|f (z)|p dvα (z) < ∞.
Bn
The space
A2α
is a Hilbert space with inner product
Z
hf, gi =
f (z)g(z) dvα (z).
Bn
1991 Mathematics Subject Classification. Primary 47B33, Secondary 32A36.
1
2
KEHE ZHU
Every holomorphic ϕ : Bn → Bn induces a composition operator
Cϕ : H(Bn ) → H(Bn ),
namely, Cϕ f = f ◦ ϕ. When n = 1, it is well known that Cϕ is always
bounded on Apα ; and Cϕ is compact on Apα if and only if
lim−
|z|→1
1 − |z|2
= 0.
1 − |ϕ(z)|2
See [2] and [3].
When n > 1, not every composition operator is bounded on Apα . For example, it can easily be checked with Taylor coifficients that the composition
operator Cϕ is not bounded on A2α when
ϕ(z) = (π(z), 0, · · · , 0),
where
π(z) =
√
nn z1 · · · zn .
See [2] for more examples and references.
The main result of the paper is the following.
Theorem. Suppose p > 0 and α > −1. If the composition operator Cϕ is
bounded on Aqβ for some q > 0 and −1 < β < α, then Cϕ is compact on
Apα if and only if
(1)
lim−
|z|→1
1 − |z|2
= 0.
1 − |ϕ(z)|2
Note that the compactness of Cϕ on Apα always implies condition (1);
we do not need any assumption on ϕ for this half of the theorem. The
assumption that Cϕ be bounded on Aqβ for some β < α is needed only for
the other half the theorem. The exponents p and q are not important.
2. P RELIMINARIES
We collect a few preliminary results in this section that will be needed
later in the paper. We begin with the notion of compact composition operators on Apα .
When p > 1, the Bergman space Apα is a reflexive Banach space (see [6]
for more information about Bergman spaces), and all reasonalble definitions
of compactness of Cϕ on Apα are equivalent. In general, for any p > 0, we
say that the composition operator Cϕ is compact on Apα if
Z
lim
|Cϕ fk |p dvα = 0
k→∞
Bn
COMPOSITION OPERATORS
3
whenever {fk } is a bounded sequence in Apα that converges to 0 uniformly
on compact subsets of Bn .
For any holomorphic ϕ : Bn → Bn we can define a positive Borel measure µϕ,α on Bn as follows. Given a Borel set E in Bn , we set
Z
−1
µϕ,α (E) = vα (ϕ (E)) = cα
(1 − |z|2 )α dv(z).
ϕ−1 (E)
Obviously, µϕ,α is the pullback measure of dvα under the map ϕ. Therefore,
we have the following change of variables formula:
Z
Z
(2)
f (ϕ) dvα =
f dµϕ,α ,
Bn
Bn
where f is either nonnegative or belongs to L1 (Bn , dµϕ,α ). In particular,
the composition operator Cϕ is bounded on Apα if and only if there exists a
constant C > 0 such that
Z
Z
p
(3)
|f | dµϕ,α ≤ C
|f |p dvα
Bn
Bn
for all f ∈ Apα . Measures satisfying this condition are called Carleson
measures for the Bergman space Apα .
Similarly, a positive Borel measure µ on Bn is called a vanishing Carleson
measure for the Bergman space Apα if
Z
|fk |p dµ = 0
(4)
lim
k→∞
Bn
whenever {fk } is a bounded sequence in Apα that converges to 0 uniformly
on compact subsets of Bn . In particular, a composition operator Cϕ is compact on Apα if and only if the pullback measure µϕ,α is a vanishing Carleson
measure for Apα .
It is well known that Carleson (and vanishing Carleson) measures for
the Bergman space Apα is indenpendent of p. More precisely, the following
result holds.
Lemma 1. Suppose p > 0 and α > −1. Then the following conditions are
equivalent for any positive Borel measure µ on Bn .
(i) µ is a Carleson measure for Apα , that is, there exists a constant C >
0 such that
Z
Z
p
|f | dµ ≤ C
|f |p dvα
Bn
for all f ∈ Apα .
Bn
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KEHE ZHU
(ii) For some (or each) R > 0 there exists a constant C > 0 (depending
on R and α but independent of a) such that
µ(D(a, R)) ≤ Cvα (D(a, R))
for all a ∈ Bn , where D(a, R) is the Bergman metric ball at a with
radius R.
Proof. See [7] for example.
A consequence of the above lemma is the following well-known result
about composition operators; see [2].
Corollary 2. Suppose p > 0, q > 0, and α > −1. Then Cϕ is bounded on
Apα if and only if Cϕ is bounded on Aqα .
A similar characterization of vanishing Carleson measures for Apα also
holds.
Lemma 3. Suppose p > 0 and α > −1. The following two conditions are
equivalent for a positive Borel measure on Bn .
(i) µ is a vanishing Carleson measure for Apα .
(ii) For some (or any) R > 0 we have
lim−
|a|→1
µ(D(a, R))
= 0.
vα (D(a, R))
Proof. See [7] for example.
As a result of the above lemma we see that the compactness of Cϕ on Apα
is indenpendent of p. We state this as the following corollary which can be
found in [2] as well.
Corollary 4. Suppose p > 0, q > 0, and α > −1. Then Cϕ is compact on
Apα if and only if Cϕ is compact on Aqα .
We need two more technical lemmas. The first of which is called Schur’s
test and concerns the boundedness of integral operators on Lp spaces. Thus
we consider a measure space (X, µ) and an integral operator
Z
(5)
T f (x) =
H(x, y)f (y) dµ(y),
X
where H is a nonnegative measurable function on X × X.
Lemma 5. Suppose that there exists a positive measurable function h on X
such that
Z
H(x, y)h(y) dµ(y) ≤ Ch(x)
X
COMPOSITION OPERATORS
5
for almost all x and
Z
H(x, y)h(x) dµ(x) ≤ Ch(y)
X
for almost all y, where C is a positive constant. Then the integral operator
T defined in (5) is bounded on L2 (X, dµ). Moreover, the norm of T on
L2 (X, dµ) is less than or equal to the constant C.
Proof. See [5] or [6].
Lemma 6. Suppose α > −1 and t > 0. Then there exists a constant C > 0
such that
Z
dvα (w)
C
≤
n+1+α+t
(1 − |z|2 )t
Bn |1 − hz, wi|
for all z ∈ Bn .
Proof. See [4].
3. C HARACTERIZATIONS IN T ERMS OF K ERNEL F UNCTIONS
In order to understand the mild assumption made in the statement of the
main theorem, we show in this section how the boundedness and compactness of composition operators on Bergman spaces can be described in terms
of Bergman type kernel functions.
Theorem 7. Suppose p > 0, α > −1, and t > 0. Then the composition
operator Cϕ is bounded on Apα if and only if
Z
dvα (z)
2 t
< ∞.
(6)
sup (1 − |a| )
n+1+α+t
a∈Bn
Bn |1 − ha, ϕ(z)i|
Proof. It follows from Lemma 6 that the boundedness of Cϕ on Apα implies
condition (6).
Next we assume that condition (6) holds. Then by the change of variables
formula (2) there exists a constant C > 0 such that
Z
dµϕ,α (z)
2 t
(1 − |a| )
≤C
n+1+α+t
Bn |1 − ha, zi|
for all a ∈ Bn . For any fixed positive radius R we have
Z
dµϕ,α (z)
2 t
(1 − |a| )
≤C
n+1+α+t
D(a,R) |1 − ha, zi|
for all a ∈ Bn . It is well known that
|1 − ha, zi| ∼ 1 − |a|2
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KEHE ZHU
for z ∈ D(a, R), and it is also well known that
(1 − |a|2 )n+1+α ∼ vα (D(a, R));
see [7]. It follows that there exists another positive constant C (independent
of a) such that
µϕ,α (D(a, R)) ≤ Cvα (D(a, R))
for all a ∈ Bn . By Lemma 1, the measure µϕ,α is Carleson for Apα , and so
the composition operator Cϕ is bounded on Apα .
This result is probably well known to experts in the field. The main point
here is that t can be an arbitrary positive constant. This also tells us roughly
how far away the boundedness of Cϕ on Apα is from that of Cϕ on Apβ .
Corollary 8. Suppose p > 0, q > 0, and −1 < β < α. If Cϕ is bounded on
Aqβ , then Cϕ is bounded on Apα .
Proof. Write α = β + with > 0. Since
(1 − |z|2 )
1 − |z|2
≤ C1
≤ C2 ,
|1 − ha, ϕ(z)i|
1 − |ϕ(z)|2
where the last inequality is an easy consequence of Schwarz lemma for the
unit ball, we have
Z
Z
(1 − |w|2 )α dv(w)
(1 − |w|2 )β dv(w)
≤
C
.
2
n+1+α+t
n+1+β+t
Bn |1 − ha, ϕ(w)i|
Bn |1 − ha, ϕ(w)i|
This shows that
2 t
Z
sup (1 − |a| )
a∈Bn
Bn
dvβ (z)
<∞
|1 − ha, ϕ(z)i|n+1+β+t
implies
Z
dvα (z)
< ∞.
n+1+α+t
a∈Bn
Bn |1 − ha, ϕ(z)i|
The desired result then follows from Theorem 7.
2 t
sup (1 − |a| )
A similar argument gives the following characterization of the compactness of Cϕ on Apα .
Theorem 9. Suppose p > 0, α > −1, and t > 0. Then Cϕ is compact on
Apα if and only if
Z
dvα (z)
2 t
(7)
lim− (1 − |a| )
= 0.
n+1+α+t
|a|→1
Bn |1 − ha, ϕ(z)i|
Corollary 10. Suppose p > 0, q > 0, and −1 < β < α. Then the compactness of Cϕ on Aqβ implies the compactness of Cϕ on Apα .
COMPOSITION OPERATORS
7
Although we do not need Hardy spaces in this paper, we mention here
that if Cϕ is bounded (or compact) on a Hardy space H q of the unit ball,
then Cϕ is bounded (or compact) on very Bergman space Apα . This result,
along with Corollaries 8 and 10 above, can be found in [1].
4. C OMPACTNESS ON B ERGMAN S PACES
We now prove the main result of the paper.
Theorem 11. Suppose p > 0 and α > −1. If Cϕ is bounded on Aqβ for
some q > 0 and −1 < β < α, then Cϕ is compact on Apα if and only if
(8)
lim−
|z|→1
1 − |z|2
= 0.
1 − |ϕ(z)|2
Proof. According to Corollary 4, we may assume that p = 2.
The normalized reproducing kernels of A2α are given by
kz (w) =
(1 − |z|2 )(n+1+α)/2
.
(1 − hw, zi)n+1+α
Each kz is a unit vector in A2α and it is clear that
lim kz (w) = 0,
|z|→1−
w ∈ Bn .
Furthermore, the convergence is uniform when w is restricted to any compact subset of Bn . A standard computation shows that
n+1+α
Z
1 − |z|2
∗
2
|Cϕ kz | dvα =
,
1 − |ϕ(z)|2
Bn
so the compactness of Cϕ on A2α (which is the same as the compactness of
Cϕ∗ on A2α ) implies condition (8).
We proceed to show that condition (8) implies the compactness of Cϕ on
2
Aα , provided that Cϕ is bounded on Aqβ for some β ∈ (−1, α). An easy
computation shows that the operator
Cϕ Cϕ∗ : A2α → A2α
admits the following integral representation:
Z
f (w) dvα (w)
∗
,
(9)
Cϕ Cϕ f (z) =
n+1+α
Bn (1 − hϕ(z), ϕ(w)i)
f ∈ A2α .
We will actually prove the compactness of Cϕ Cϕ∗ on A2α , which is equivalent
to the compactness of Cϕ on A2α . In fact, our arguments will prove the
8
KEHE ZHU
compactness of the following integral operator on L2 (Bn , dvα ):
Z
f (w) dvα (w)
(10)
T f (z) =
.
n+1+α
Bn |1 − hϕ(z), ϕ(w)i|
For any r ∈ (0, 1) let χr denote the characteristic function of the set
{z ∈ Cn : r < |z| < 1}. Consider the following integral operator on
L2 (Bn , dvα ):
Z
(11)
Tr f (z) =
Hr (z, w)f (w) dvα (w),
Bn
where
χr (z)χr (w)
|1 − hϕ(z), ϕ(w)i|n+1+α
is a nonnegative integral kernel. We are going to estimate the norm of Tr on
L2 (Bn , dvα ) in terms of the quantity
Hr (z, w) =
1 − |z|2
.
2
r<|z|<1 1 − |ϕ(z)|
Mr = sup
We do this with the help of Schur’s test.
Let α = β + σ, where σ > 0, and consider the function
h(z) = (1 − |z|2 )−σ ,
z ∈ Bn .
We have
Z
Z
cα
χr (z)χr (w) dvβ (w)
Hr (z, w)h(w) dvα (w) =
cβ Bn |1 − hϕ(z), ϕ(w)i|n+1+β+σ
Bn
Z
χr (z) dvβ (w)
cα
≤
.
cβ Bn |1 − hϕ(z), ϕ(w)i|n+1+β+σ
By the boundedness of Cϕ on Aqβ , there exists a constant C1 > 0, independent of r and z, such that
Z
Z
dvβ (w)
Hr (z, w)h(w) dvα (w) ≤ C1 χr (z)
.
n+1+β+σ
Bn
Bn |1 − hϕ(z), wi|
We apply Lemma 6 to find another positive constant C2 , independent of r
and z, such that
Z
C2 χr (z)
Hr (z, w)h(w) dvα (w) ≤
(1 − |ϕ(z)|2 )σ
Bn
σ
1 − |z|2
= C2 χr (z)
h(z)
1 − |ϕ(z)|2
≤ C2 Mrσ h(z)
COMPOSITION OPERATORS
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for all z ∈ Bn . By the symmetry of Hr (z, w), we also have
Z
Hr (z, w)h(z) dvα (z) ≤ C2 Mrσ h(w)
Bn
for all w ∈ Bn . It follows from Lemma 5 that the operator Tr is bounded
on L2 (Bn , dvα ) and the norm of Tr on L2 (Bn , dvα ) does not exceed the
constant C2 Mrσ .
Now fix some r ∈ (0, 1) and fix a bounded sequence {fk } in A2α that
converges to 0 uniformly on every compact subset of Bn . In particular,
{fk } converges to 0 uniformly on |z| ≤ r. We use (9) to write
Cϕ Cϕ∗ fk (z) = Fk (z) + Gk (z),
z ∈ Bn ,
where
Z
Fk (z) =
|w|≤r
fk (w) dvα (w)
,
(1 − hϕ(z), ϕ(w)i)n+1+α
and
Z
χr (w)fk (w) dvα (w)
.
n+1+α
Bn (1 − hϕ(z), ϕ(w)i)
Since {fk (w)} converges to 0 uniformly on for |w| ≤ r, we have
Z
lim
|Fk (z)|2 dvα (z) = 0.
Gk (z) =
k→∞
Bn
For any fixed z ∈ Bn , the weak convergence of {fk } to 0 in L2 (Bn , dvα )
implies that Gk (z) → 0 as k → ∞. In fact, by splitting the ball into |z| ≤ δ
and δ < |z| < 1, it is easy to show that
lim Gk (z) = 0
k→∞
uniformly for z in any compact subset of Bn .
It follows from the definition of Tr that
Z
Z
Z
2
2
|Gk | dvα ≤
|Gk | dvα +
|z|≤r
Bn
|Tr (|fk |)|2 dvα .
Bn
Since {fk } is bounded in L2 (Bn , dvα ), and since the norm of the operator
Tr on L2 (Bn , dvα ) does not exceed C2 Mrσ , we can find a constant C3 > 0,
independent of r and k, such that
Z
|Tr (|fk |)|2 dvα ≤ C3 Mr2σ
Bn
for all k. Combining this with
Z
lim
k→∞
|z|≤r
|Gk |2 dvα = 0,
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KEHE ZHU
we obtain
Z
|Gk |2 dvα ≤ C3 Mr2σ .
lim sup
k→∞
Bn
This along with the estimates for Fk in the previous paragraph gives
Z
lim sup
|Cϕ Cϕ∗ fk |2 dvα ≤ C3 Mr2σ .
k→∞
Bn
Since r is arbitrary and Mr → 0 as r → 1− (which is equivalent to the
condition in (8)), we conclude that
Z
lim
|Cϕ Cϕ∗ fk |2 dvα = 0.
k→∞
Bn
So Cϕ is compact on A2α , and the proof of the theorem is complete.
Note that when n = 1, Cϕ is bounded on every Bergman space Aqβ , so the
characterization of compact composition operators on Apα does not need any
extra assumption. However, our proof here still works. The idea of using
Schur’s test to prove the compactness of composition operators seems to be
new even in the case n = 1.
A similar compactness result was proven in [1] for composition operators
on Apα . But the condition in [1] involves the derivatives of ϕ and is much
stronger than our condition here.
Finally we mention that our results and proofs generalize to strongly
pseudo-convex domains. The key preliminary results we need are Lemmas 1, 3, 5, and 6, which are all known to be true for strongly pseudoconvex domains. For the same reasons, our results and proofs also work for
products of balls in Cn . In particular, they remain valid for the polydisk.
R EFERENCES
[1] D. D. Clahane, Compact composition operators on weighted Bergman spaces of the
unit ball, J. Operator Theory 45 (2001), 335-355.
[2] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions,
CRC Press, Boca Raton, 1994.
[3] B. MacCluer and J. Shapiro, Angular derivatives and compact composition operators
on the Hardy and Bergman spaces, Canadian J. Math. 38 (1986), 878-906.
[4] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980.
[5] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.
[6] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York,
2004.
[7] K. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), 329-357.
D EPARTMENT OF M ATHEMATICS , SUNY, A LBANY, NY 12222, USA
COMPOSITION OPERATORS
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AND D EPARTMENT OF M ATHEMATICS , S HANTOU U NIVERSITY , S HANTOU , C HINA
E-mail address: [email protected]